Abstract: We used the cubic B-spline finite element method to solve a class of fourth order parabolic equations with variable coefficient, and proved the boundness and convergence of the solution of the semi-discrete scheme. For the discretization of time variable, the backward Euler scheme was
 constructed, and the convergence order of the solution of the fully discrete scheme was O(Δt+h⁴). A numerical experiment verified the effectiveness of theoretical analysis results and the B-spline finite element method.

Key words: variable coefficient, fourth order parabolic equation, B-spline, finite element method